RM - Instrumental Variables

OLS estimator (formula words)

causal + selection

beta OLS

beta + Cov(e,D) / Var(D)

Selection bias in OLS estimator (formula)

COV(e,D) / Var (D)

OLS estimator if we have imperfect randomization

does not reflect the causal effect of taking the treatment but is biased by a selection term

Endogeneity (words)

the covariance of the treatment and the error term is not zero = OLS problematic

Exogenous (condition + OLS)

Covariance between the treatment and the error term is 0 = OLS no problem

IV method

uses three assumptions to characterise a chain reaction leading from the instrument to outcome

Three IV assumptions

1) Relevance of the instrument
2) independence assumption
3) exclusion restriction

IV Assumption 1 - Relevance

the instrument is correlated with the treatment
COV(Z, D) is unequal 0

IV assumption 2 - Independence

The instrument is not correlated with errors

IV assumption 3 - Exclusion restriction

The instrument only affects the outcome Y through the treatment D

Testing IV assumption

1) can be directly tested via regression from data
2) & 3) cannot be directly tested, the researcher needs to argue why this holds

Instrument is strong enough when

f-statistic of first regression > 10

OLS estimator (formula)

COV(D,Y) / Var(D)

IV estimator OLS (formula)

COV (Z,Y) /COV (Z,D)

IV estimator (words)

ratio of two OLS estimators (rho and phi)

Phi (formula)

COV(Z,D) /Var (Z)

Phi (words)

Covariance of Instrument and Treatment divided by variance of Instrument

Rho (formula) - (co)variances

COV (Z,Y) / Var (Z)

Rho (words)

Covariance of the Instrument and Outcome divide by Variance of Instrument

Rho also called

reduced form
intent to treat

Phi also called

first stage

to get causal effect via manual IV

estimate both regressions (reduced and first stage) and divide the reduced by the first one

RHO written as Expectations (formula)

E[Y | Z = 1] - E[Y | Z = 0] 

Phi written as expectations (formula)

E [ D | Z = 1] - E[ D| Z = 0] 

RHO in terms of expectations (words)

The expected outcome of those who have the instrument MINUS the expected outcome of those without the instrument

PHI in terms of expectations (words)

The likelihood of getting the treatment when having the instrument MINUS the likelihood of getting the treatment when NOT having the instrument

Why do we need a strong first stage?

so we do not divide by zero

How does IV get rid of the selection bias?

BECAUSE of the assumption that the instrument is not correlated with the error (the thing causing selection bias)

How does the 2SLS get rid of selection bias?

using the fitted values for the treatment gets rid of the error (because fitted values are predicted values, not real ones = no error)

PHI when we have perfect compliance

is 1

When Phi < 1

we are dealing with incomplete compliance
= the Intent to Treat is bigger than the local average treatment effect

LATE formula

gamma = rho / phi

Why are we estimating only the effect of compliers with IV?

because we only have variation in the treatment (e.g. we only observe treatment on those who otherwise would not get it)

never takers

those who would never take the treatment

always takers

those who would always take the treatment

Assumption needed to interpret IV beta as LATE

Monotonicity assumption

Monotonicity assumption

there are no defiers

Defiers

those who would only take treatment if they do not get the instrument (randomization)

Two-stage least squares

alternative way of computing ratio of rho and phi by means of a sequence of two regressions

Regression in the two-stage least squares

1) Regress the instrument on the treatment variable
2) save the predicted values from Regression 1 and regress the outcome variable on these predicted values

Advantages of 2SLS

1) allows as many control variables as you’d like
2) allows to efficiently use more than 1 instrument per exogenous covariate

Steps to 2SLS

1) test instrument
2) estimate the impact of instrument on treatment
3) Use fitted values of treatment for second-stage equation

2SLS - Regression 0

Yi = a0 + rho * Zi + y0 Ai + e0i

2SLS - Regression 1 (formula)

Di = a1 + Phi * Zi + y1Ai + e1i

2SLS - Regression 2

Yi = a2 + beta(2SLS) D^HAT + y2Ai + e2i

Parameters in 2SLS

rho = overall effect of instrument on outcome
phi = effect of instrument on treatment
beta = true causal effect of treatment on outcome
ei = confounder that affects relationship of treatment and outcome